Introductory Biostatistics

DATA;

INPUT GROUP $ COTININE;

DATALINES;

U8

...

U 111

E35

...

E 208;

PROC NPAR1WAY WILCOXON; CLASS GROUP;

VAR COTININE;

7.4.2 Wilcoxon Signed-Rank Test

The idea of usingranks, instead of measured values, to form statistical tests to
compare population means applies to the analysis of pair-matched data as well.
As with the one-samplettest for pair-matched data, we begin by forming dif-
ferences. Then the absolute values of the di¤erences are assigned ranks; if there
are ties in the di¤erences, the average of the appropriate ranks is assigned.
Next, we attach aþor a
sign back to each rank, depending on whether the
corresponding di¤erence is positive or negative. This is achieved by multiplying
each rank byþ1,
1, or 0 as the corresponding di¤erence is positive, negative,
or zero. The results aren signed ranks, one for each pair of observations; for
example, if the di¤erence is zero, its signed rank is zero. The basic idea is that if
themean di¤erenceis positive, there would be more and largerpositive signed
ranks; since if this were the case, most di¤erences would be positive and larger
in magnitude than the few negative di¤erences, most of the ranks, especially the
larger ones, would then be positively signed. In other words, we can base the
test on thesum Rof thepositive signed ranks. We test the null hypothesis of no
di¤erence by calculating thestandardized test statistic:

z¼

R
mR

sR

where

mR¼

ðnÞðnþ 1 Þ

4

is the mean and

sR¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðnÞðnþ 1 Þð 2 nþ 1 Þ

24

r

NONPARAMETRIC METHODS 261